Such as in the LiveJournal social network, population density is non-uniform in real social networks[1]}. To deal with the navigation problem with non-uniform population density, a scale invariant friendship network (SIFN for short) model based on the above spatial scaling law \(P(d)\propto d^{-1}\)  of social networks is proposed in this paper. We prove the time complexity of navigation in a 2-dimensional SIFN is at most \(O(\log ^4 n)\) , which indicates social networks is navigable. Dodds et
al have pointed out that individuals often resort to extra information such as education and professional information besides geography location in the real searching experiment[2]}. Considering this phenomenon, navigation process in real world may be seen as the projection of navigation in a higher dimensional SIFN. Therefore, we further discuss the relationship between high and low dimensional SIFN. Particularly, we prove that a 2-dimensional SIFN can be seen as the projection of any \(k\) -dimensional SIFN(\(k>2\) ) through theoretical analysis.
